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NBER WORKING PAPER SERIES EXPORT SUPPLY AND IMPORT DEMAND FUNCTIONS: A PRODUCTION THEORY APPROACH W. Erwin Diewert Catherine J. Morrison Working Paper No. 2011 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 August 1986 Research support from the National Science Foundation, Grant SES-8420937, is gratefully acknowledged. The research reported here is part of the NBER's research programs in International Studies and Productivity. Any opinions expressed are those of the authors and not those of the National Bureau of Economic Research. NBER Working Paper #2011 2\ugust 1986 Export Supply and Import Demand Functions: A Production Theory Approach ABSTRACT In this paper we theoretically and empirically model import demand and export supply behavior of firms for the U.S. economy from 1967-1982. A producer theoretic approach based on duality theory is used to derive econometric systems of producer supply and demand functions that are consistent with profit maximizing behavior. This syste. is then empirically l.plemented and the resulting estimates used to construct a full set of supply and demand elasticities characterizing import demand and export supply functions as well as domestic output supply and labor demand. These elasticities are in turn used to derive devaluation elasticities and some estimates of the equilibrium real exchange rate that would cause the U.S. trade surplus to reach zero. W. Erwin Diewert Department of Economics University of British Columbia Vancouver, B.C., Canada V6T 1Y2 Catherine J. Morrison Department of Economics Tufts University Medford, Massachusetts 02155 I I. Introduction In this paper, we utilize a producer theory approach to the generation of export supply and import demand functions for the U.S. economy for the period 1967 to 1982. Our approach uses an economy wide GNP function1 or restricted profit function with exports appearing as outputs and imports appearing as inputs. This approach to modeling trade functions using an integrated production theory framework was implemented by Kohli [19781 for the Canadian economy.2 Traditional general equilibrium models of trading economies assume that internationally traded goods can enter the consumer sector directly and hence traditional empirical approaches to the estimation of import demand and export supply functions inevitably involve modeling the household sector. However, in Kohlis approach, imports are regarded as intermediate inputs into the production sector and exports are regarded as separate nondomestic outputs produced by the private production sector as a whole.3 Treating imported goods as intermediate goods seems reasonable since: (i) many imports are intermediates and (ii) even imported goods that are delivered to the household sector generally have domestic transportation, wholesaling and retailing inputs added to them. The advantage of the Kohli approach over the traditional approach is that we need only model the private production sector of the economy and we can ignore the difficult problems involved in modeling the consumer sector. Duality theory may be used to derive econometrically convenient systems of producer supply and demand functions that are consistent with profit maximizing behavior.4 However, often the estimated profit functions do not 2 satisfy the appropriate theoretical curvature conditions. Thus in this study, we shall use an adaptation to the profit function context of a functional form due to Diewert and Wales [1985]. This functional form allows one to impse the correct curvature conditions in a global manner without destroying the flexibility of the functional form. A brief outline of the paper follows. In Section 2, we outline the production theory approach to the generation of systems of export supply and import demand functions. In Section 3, we introduce our functional form for the profit function and we explain how the correct curvature conditions can be imposed. Empirical results are also presented in this section. In Section 4, we table various elasticities based on our estimated profit function. In Section 5, we derive some devaluation elasticities and some estimates of the equilibrium real exchange rate by year that would drive the U.S. merchandise trade surplus or deficit to zero for the years 1967 to 1982. For example, for the year 1982, we estimate that a devaluation of 28% would be required to eliminate the U.S. balance of trade deficit for that year. Section 6 concludes and our data are described and tabled in a data Appendix. Our model of producer behavior is based on short run competitive profit maximization, holding capital fixed in each period. Thus we do not model the capital accumulation process. Moreover, we assume that producers regard wage rates as fixed and can hire any amount of labor at the going wage rate. Due to the existence of substantial amounts of unemployment during the period, we feel that this assumption is •ore realistic than the polar case where labor input into the private production sector is held fixed and the wage rate becomes an endogenous variable. In subsequent work, we plan to consider alternative treatments of labor. 3 2. A Prod uction The O!1 Ap2r oa ch t oEortp2l_yandIportDemandFunctions We assume that domestic and international demand and supply decisions are made by profit maximizing firms operating under perfect competition in all commodity and factor markets. The firms face vectors of domestic and international prices and a vector of domestic primary factor stocks and make decisions about domestic and international input demand and production. The technology is assumed to be represented by a production possibilities set, from which a profit function can be derived. Specifically, let the production possibilities set for the private sector of a country in period t be a set St{(x,y,Kt)} where x=(x1 xN) is an N dimensional vector of domestic net outputs (if x<O, then good n is an input), y(y1,. . .,yj) is an M dimensional vector of net exports (if Ym<O, then good in is an imported good that is being utilized by the private sector) and KtEKt1,.. .,Ktj) is a nonnegative J dimensional vector of capital stocks utilized by the private sector in period t. Let ptpt1 pt) be the positive vector of domestic prices faced by domestic producers in period t and let wt=(wt1... ,wtM) be the positive vector of international prices that producers face in period t. Then we can define the country's net revenue or restricted profit function (see Gorman [1968], McFadden [1966], or Diewert [1974fl by: t t t t r (p ,w ,K ) (1) flax y t t t (p .x + w .y : (x,y,K ) t cS ) From I1otelling's Lemma, the economy's observed net output vector x = (xtj...,xtj) and observed net export vector (yt1,...yt) are equal to the vectors of derivatives of the restricted profit function with respect to the components of Pt and wt, respectively (if the derivatives exist); i.e., 4 tttt (2) xt = Vu (3) t ' = (p ,w ,K ) and t t ,wt,Kt) (p for t=1 T, where V is the vector differential operator. Thus Vflt(pt,wtKt) is the vector of first order derivatives of n with respect to the components of p. Note that these net output and input vectors are explicitly based on short run decisions since they are conditional on fixed stock levels of certain inputs. The extent of fixity —- and thus the flexibility of the country to act in the short run —- will depend on how many stocks are held fixed (e.g., recognizing only the fixity of capital plant and equipment, or including also labor stocks to characterize labor hoarding, or assuming a given domestic output level which generates the negative of a cost function). Assuming that only capital stocks are fixed, the system of equations (2) represents the economy's short run domestic output supply and labor (and possibly other domestic variable input) demand functions while the system of equations (3) contains the economy's short run export supply and import demand functions. If we assume a functional form for fl, and append errors to (2) and (3), then the unknown parameters which characterize flt (and therefore the production possibilities set St by duality theory) may be econometrically estimated from (2) and (3), given a time series of observations on xt, yt, Pt, wt, and Kt. 3. The_Normalized_Quadratic Prof it Function The approach explained in and the previous section for modeling export supply import demand functions has been implemented by Kohli [1978] for Canada using the translog functional form. However, a problem with the translog 5 functional form is that the estimated profit function frequently fails to satisfy the appropriate theoretical curvative conditions. A procedure due to Jorgenson and Fraumeni [19811 (see also Jorgenson [19841) may be used to impose the appropriate curvature conditions, but Diewert and Wales [1985] have shown that this procedure destroys the flexibility of the translog functional form. We therefore will implement the model described in section 2 by using a special case of the Biquadratic Restricted Profit Function defined by Diewert [1985;89]. For the case of only one capital good and a constant returns to scale technology, this profit function in period t, flt(p,w,K) say, is defined as follows: t n (4) t (p,w,K t )IK [p + TT ,w Ia + [pTT ,w Jbt TT T (1/2)[p2,...,PN;w IB[p2....,pN;w II /p1 where Kt>O denotes the quantity of capital utilized during year t, t is a scalar indicator of technical progress In time period t, p and w [w1 [p1,p2,...,pIT g.]T are N and M dimensional (column) vectors of domestic and international prices respectively, a [al,...,aN+M]T and b bN+MIT [b1 are Ni-K dimensional column vectors of unknown parameters and B [b1] is an N-l-i-M by N-li-K symmetric matrix of unknown parameters. Note that the right hand side of (4) defines a unit profit function: it defines the maximum amount of net revenue that the economy can produce, given one unit of capital and given that all producers face the domestic prices p and the international prices w. Note that our definition of flt has imposed a constant returns to scale property on the technology: if we change the capital input by the scalar )..>O, then the economy's net revenue will change by the same proportional factor X; i.e., flt(p,w,Xkt) = Xflt(p,w,Kt). The function becomes more complex, of course, if more fixed inputs are considered. 6 The profit function defined by (4) is a straightforward adaptation of the Generalized McFadden cost function defined in Diewert and Wales L1985]. Adapting their proofs, it can be shown that the Ut defined by (4) is a flexible functional form for a constant returns to scale technology. It can be seen the n also is linearly homogeneous in prices, although it will be convex in prices if and only if the symmetric matrix B is positive semidefinite. For our empirical work, we utilize the data for the U.S. economy for the period 1967-1982 which was developed by Morrison and Diewert [19851. We have two domestic goods (i.e., N=2): x1 is the quantity of domestic sales and x2 is (minus) labor input.5 We have three classes of internationally traded goods: Yi is (minus) the quantity of import demand excluding petroleum products, Y2 is (minus) petroleum imports, and Y3 is exports. The data are more fully described in the data Appendix. From (2) and (3), differentiating fl with respect to the prices p1,p2,w1,w2,w3 yields: (I) the domestic supply function, (ii) (minus) the labor demand function, (iii) (minus) the nonpetroleum import demand function, (iv) (minus) the petroleum import demand function and (v) the export supply function for the U.S. economy in period t. We divided these net supply functions by the capital utilized in the period (this made the assumption of homoskedastic errors more plausible) and appended the stochastic disturbance ut1 to equation i in period t. The resulting system of estimating equations is given by 7 (5) - — — (6) (7) (8) (9) + x/Kt = a1 t t x2/K = a2 t t y2/K — (l/2)b11(p/p)2 t t2 (112)b33(w2/p1) tt t2 b13p2w2/(p1) tt t2 b24w1w3/(p1) + t b2t t t yr/K = a3 b1tt i- b3t t t a4 + b4t t t t ye/K = a5 + b5t + + + + — — — (l/2)b22(w/p)2 t t2 — tt t2 b14p2w3/(p1) tt t2 t t b12p2/p1 t t b13p2/p1 t t b14p2/p1 + + + + t t b12w1/p1 t t b22w1/p1 t t b23w1/p1 t t b24w1/p1 tt t2 b12p2w1/(p1) tt t2 — b23w1w2/(p1) t u + b34w2w3/(p1) t t b11p2/p1 — (1/2)b44(w3/p1) + + + + t t b13w2/p1 t t b23w2/p1 t t b33w2/p1 t t b34w2/p1 + + + + t t b14w3/p1 t t b24w3/p1 t t b34w3/p1 t t b44w3/p1 t + 112 t + + + t u4 t u5 There are five a1, five bi and ten b1 parameters that must be estimated given that the cross equation symmetry constraints on the b1 in equations (5)—(9) have been imposed. To estimate this system, we first assume that the error vectors jtE[ut11.. 11t5jT are independently distributed with a multivariate normal distribution with zero means and covariance matrix 0. We then were able to obtain the maximum likelihood estimates for a,b and B using the iterative Zeliner SYSTEM command in version 5 of SHAZAM (see White [1978)) because the system is linear. RESTRICT statements were used to impose the cross equation symmetry restrictions. The algorithm took 65 iterations to converge. The log of the likelihood function after one iteration was 210.25 and after 65 iterations was 260.04. The equation R-squares between the observed and predicted values were: .24,.80,.45,. 25, and .93. These R— squares are relatively low because of our division of the quantities on the 8 left hand side of (5)-(9) by capital to eliminate trends due to the growth in the economy. Our parameter estimates may be found in Table 1 below. Unfortunately, this unconstrained specification resulted in a violation of the required curvature properties for the technology. Testing for this violation requires verifying that the matrix of second order partial derivatives of II with respect to prices, V2zzrt(zt,Kt) is positive semidefinite. Since the price by definition is positive and the functional form is basically a quadratic, the convexity conditions are satisfied globally if and if the estimated B matrix is positive semidefinite, which can be determined by checking whether the elgenvalues are all positive. For our estimated unconstrained model, one of these elgenvalues turned out to be negative. Since the estimated B matrix did not satisfy the required properties we imposed positive semi-definiteness on B by a reparametrization due to Wiley, Schmidt and Bramble [1973] and discussed and extended in Diewert [1985] and Diewert and Wales [1985]. The process requires replacing the components of B by the corresponding elements of a 4 by 4 lower triangular matrix C, where 10) B=CCT; C [c..]; i,j = 1,2,3,4; c13O for j>i Due to the symmetry restrictions imposed on the b3 parameters, this reparameterization does not change the total number of parameters to be estimated. However the equations become substantially more complicated since the b1 parameters, which enter linearly in (5)—(9) given the restrictions on the model, must be replaced by 11) b.. = c.kc.k for i,j=1,. . ,' 9 Using equations (10) or (11), the b1j's appearing in (5)-(9) were replaced by these functions of the Cjj'S. and the resulting parameters ai,bj, and C were estimated with the nonlinear regression command in SHAZAM using the Davisdon=Fletcher—Powell algorithm which required 345 iterations to converge. At the final stage, the log of the likelihood function was 257.8 and the equation by equation k—squares between the observed and predicted values were: .23, .79, .37, .25 and .91. Thus the likelihood decreased negligibly going from the B unconstrained case to the B constrained case and there was little loss of fit. Once estimates for the elements of the C matrix have been found, estimates for the elements of the B matrix can be obtained using matrix equation (10), or, equivalently, equations (11). These estimates for the b13 parameters along with the corresponding estimates for the a1 and bjts are reported in the last column of Table 1. Standard errors are not reported since standard asymptotic theory is unfortunately not applicable when parameters are subject to inequality constraints.6 As the reader can see from Table 1, in most cases the constrained parameter estimates are of the same sign and order of magnitude as the corresponding unconstrained case. In the following sections, we use only the constrained estimates in order to form a system of U.S. net domestic supply functions by differentiating our estimated profit function with respect to domestic prices (recall (2)) and a system of U.S. net export supply functions by differentiating with respect to international prices (recall (3)). 10 4. Trade Elasticities We differentiate nt(pt1,pt2,wt1,wt2,wt3,Kt) with respect to p1 and P2 to obtain a domestic supply function xt1 flt/pj and (minus) a labor demand function x2 t/p2. Similarly, differentiation of fl with respect to w2, and w3 yields ytj = function, yt2 t/w1 = (minus) a nonpetroleum import demand flt/w2 = (minus> a petroleum products import demand function and yt3 t/q3 = an export supply function. It is of some interest to consider the various supply and demand elasticities with respect to prices.7 In Table 2 below, we list the domestic supply elasticities Ex1p = (wtj/xt1).(xt1/w), (pt1/xt1)(xt1/p1), i=1,2, and Ex1w3 = j=1,2,3 for alternate years. In tables 3—6 we list the price elasticities for the labor demand, nonpetroleum import demand, petroleum import demand and export supply functions, respectively. From Table 2, it can be seen that in 1982, a 1% increase in the domestic price level or the price of nonpetroleum imports would increase domestic supply (holding capital and other prices fixed) by about .8% and .01% respectively while a 1% increase in the wage rate, the price of exports or the price of petroleum imports would decrease domestic supply by about .7%, .1% and .1% respectively. It is interesting to note that the one sign change appearing over time in these tables is for petroleum imports; until 1973 a 1% increase in the price of petroleum imports corresponded to an increase in domestic supply and later it caused a decrease in supply. This has important implications about the impact of the OPEC price shocks on domestic production. In the cost function context, two inputs are Hicks [1946] -Allen [1938;5041 substitutes (complements) if the cross partial derivative of the cost function with respect to the prices of the two goods is positive 11 (nonpositive). In the profit function case, we define two distinct goods to be substitutes (complements) if the cross partial derivative of the profit function with respect to the two prices is gative (nonnegative). Hicks [1946;311-312] shows that in the two input case, the two inputs must be substitutes while in the three input case, at least two pairs of inputs must be substitutes. Hicks [1946;321] and Diewert [l974;143—1451 show more generally that in the case of only two variable goods, the two goods must be substitutes while in the three variable good case, at least two pairs of goods must be substitutes. Since the elasticities in Table 2 are second order derivatives of the profit function with respect to Pi and one of the other prices times the other price divided by xt1, it can be seen that domestic sales and exports are substitutes, which we might expect. Domestic sales and labor are substitutes while sales with either of the two types of imports are complementary pairs of goods. From Table 3, it can be seen that in 1982, a 1% increase in the domestic price level, the price of imports or the price of petroleum imports would increase the demand for labor by about 1.1%, .1%, and .02% respectively, while a 1% increase in the wage rate or in the price of exports would decrease labor demand by about 1.2% and .1% respectively. It appears that labor is quite highly substitutable with domestic sales, and slightly substitutable with both nonpetroleun and petroleum imports. Labor is complementary with exports. The relatively large own price elasticity of demand for labor means that there would appear to be a rather high wage—unemployment tradeoff; i.e., moderation in wage demands would lead to relatively large employment Increases. Conversely, the small elasticity with respect to petroleum imports implies that there is very little tradeoff between use of energy and labor in production. 12 Turning to Table 4, in 1982 a 1% increase in the wage rate or the price of petroleum imports would increase the demand for nonpetroleum imports by about 1% and .09* respectively, while a 1% increase in the domestic price level, the price of nonpetroleum imports or the price of exports would decrease the demand for nonpetroleum imports by about .2%, .7%, and .2%. Note that the own price elasticity of demand for nonpetroleum imports was close to -1 for much of the period. It can be seen that nonpetroleum imports are substitutable with labor (as might be expected) and petroleum imports, and are complementary with domestic sales and exports. Note that the substitutability with labor is particularly strong. From Table 5, in 1982 a 1% increase in the wage rate or in the price of nonpetroleum imports would increase the demand for petroleum imports by about .3% and .2% respectively, while a 1% increase in the domestic price level, the price of petroleum imports or the price of exports would decrease the demand for petroleum imports by about .25%, .6%, and .1% respectively. It can be seen that petroleum imports are substitutable with labor and nonpetroleum imports, and are complementary with domestic sales and exports. Note that although all elasticities are small, the complementarity with sales is relatively strong in the first part of the sample, decreases, and then rises again approximately to its initial value, whereas the substitutability with labor is larger at the beginning of the period (Ey2p2 diminishes consistently over time (Ey2p2 .414 in 1967) and .965 in 1982). Conversely, the own elasticity begins very small (Ey2w2 = — .182 in 1967) and increases over time, with a noticeable jump post 1973 and a peak in 1980 of — .894 about the time of the second OPEC shock. Finally, turning to Table 6, we see that in 1982 a 1% increase in the wage rate, in the price of nonpetroleum imports, in the price of petroleum 13 imports or in the price of exports would increase the supply of exports by about .8%, .2%, .07%, and .3% respectively while a 1% increase in the price of domestic sales (holding other prices and capital input constant as usual) would decrease the supply of exports by about 1.4%. Thus exports are strongly substitutable with domestic sales, are quite strongly complementary with labor and are weakly complementary with the two classes of imports. The own price elasticity of export supply stayed fairly constant between .32 and .375 over the sample period. Although these elasticities are interesting in their own right, it may be even more important to consider their combined implications for questions that we may ask about international trade. One very interesting question which may be posed, for example, is how large a change in the real exchange rate would be required to create a zero trade surplus. We turn now to an illustration of this application of the above elasticities. 5. Devaluation_and the Balance of Trade Let e be an exchange rate and define the balance of trade function In period t, Bt(e), as the value of exports minus the value of imports expressed in world prices.8 Since net export supply functions can be obtained by differentiating the profit function with respect to the international prices Wj (recall (3), Hotellings Lemma), we may define Bt(e) as follows: t (12) 8 (e) E 3 t t E.1 w.n t t t t t t (ep1,ep2,w1,w2,w3,K )/w. Since e is a hypothetical real exchange rate which expresses the price of nontraded U.S. domestic goods relative to internally traded goods, a decline in e implies a devaluation of the U.S. dollar.9 If we define it by (4) using 14 our constrained parameter estimates, it can be seen that Bt(l) corresponds to the estimated trade balance for the U.S. for the year t; Bt(1)>O (<0) indicates that the estimated trade balance is a surplus (deficit). The Bt(1) are tabled in Table 7 below. Differentiate (12) with respect to e and evaluate the resulting derivatives at e=1: (13) Bt(l) = E1 w1E1 Pt 2nt(ptwtKt)/wP where yt(p,w,K) is the jth net export supply function for the U.S. economy. Since Bt(1)0 in our sample, we can convert Bt(1) into an elasticity by multiplying it by 1/Bt(1): Bt(l)/Bt(l) = (14) = where S3 = j E.1 wy1 3 2 ti....1 =1 (pt/yt)(yt(ptwtKt)/p)/Bt(l) t t s.Ey.p. wtyt)/Bt(l) for j=1,2,3. Note that 33=1 is the period t price elasticity of 53 = 1 and Ey3pt1 with respect to Pj (see Tables 4, 5, and 6 for a partial listing for these elasticities). Equations (12) and (14) enable us to compute Bt'. A linear approximation to Bt(e) is: (15) Bt(e) Bt(l) + Bt(l)(e_l) From (15), it can be seen that a one cent appreciation in the real exchange rate in period t will change the balance of trade net surplus by approximately •0l8t'(1) billions of dollars (all quantities are measured in billions of 1972 dollars and all prices are set equal to 1 in 1972). These numbers are tabled in Table 7. It can be seen that these trade_balance impacts are all negative as we might expect;1° a devaluation of domestic as compared to traded goods 15 always improves the U.S. balance of trade. Note that the effectiveness of a hypothetical devaluation improves over the sample period 11: a one cent devaluation in the late sixties would tend to improve the trade balance by only .5 billion dollars while the same devaluation in 1981 would improve the balance by about 4 billion dollars. If we take the right hand side of (15) and set it equal to zero, we may solve for an approximate qilibrium reaL exchang rate that would induce a zero trade surplus in each year. The resulting series et may also be found in Table 7. Note that e1982 = .72, so that an approximate 28% devaluation would be required to eliminate the estimated 1982 trade deficit. However, in addition to the crudeness of the approximation (15), the reader should note that our model assumes that domestic prices are "sticky" in the short run so that the relative price of output and labor remain unchanged. In the long run, domestic demand conditions may alter these prices and hence our equilibrium exchange rate may be only a short run equilibrium. In addition, moving to an analysis of long run adjustment is much more complex because the dynamics of investment and different capital vintages should be incorporated in order to model lags associated with real world devaluations. Note, finally, that the model of real exchange rate devaluation or appreciation developed here can be adapted to yield a producer oriented theory of purchasing power parities12: an equilibrium PPP price vector for a country in period t could be defined to be the price vector etpt where the scalar exchange rate et) solves the following equation: t t tt t t (16) w •V fl(e p ,w ,K ) = T where w is the vector of international prices that all countries face, is the country's net export vector and T denotes the long run equilibrium 16 level of "transfers" (i.e., remittances abroad, net tourist expenditures abroad, net international debt service,'3 etc.) that the production sector of the country should make in period t. If labor were treated as a fixed good so that it appears in the "capital" vector Kt, and there were only one domestic good, then the scalar etpt would be the country's equilibrium PPP. This formulation is useful because it shows that a country's equilibrium PPP will not generally be unity even if technology sets are identical across countries. The equilibrium PPP depends on: (i) the country's long run level of net transfers T, (ii) the country's technological capabilities (i.e., the production set St or its dual nt), (iii) the world price vector w, and (iv) the country's endowments of labor skills, capital stocks and natural resources that are embodied in the vector Kt. Changes in any of these variables will generally change the country's equilibrium PPP. 6. Conclusion We have shown how a consistent framework for simultaneously modeling a country's domestic supply, labor demand, import demand and export supply functions can be obtained using producer theory. We followed in the footsteps of Kohli [1978] and others, except that we adapted to the profit function context some of the recent work by Diewert and Wales [1985] on flexible functional forms that satisfy curvature conditions globally. Use of the producer—oriented framework to model export supply and import demand functions is much simpler than traditional general equilibrium approaches since we do not have to model the consumer side of the model and the associated aggregation over consumers problem. However, consideration of only one "building block" of the full general equilibrium model, as with any applied production theory analysis, requires partial equilibrium assumptions such as 17 exogeneity of domestic prices rather than allowing these prices to be affected by changes in international prices. Our partial equilibrium framework requires assumptions about conditional distributions which may not be true and thus our estimates may be subject to some simultaneous equations bias. Additional problems and deficiencies arise from the simplifications incorporated in our model including: (i) a high level of aggregation, (ii) the assumption of constant returns to scale, (iii) the assumption of price taking behavior (in particular, possible induced changes in foreign prices are ignored and possible monopolistic behavior is assumed away), (iv) the neglect of natural resource stocks, (v) the use of the assumption of homogeneous capital across time periods (in particular, we have not allowed for possible adjustment costs involved in installing new capital equipment), and (vi) avoidance of consideration of the various taxes, tariffs, subsidies and quotas that impact the allocation of resources within the private production sector of the U.S. economy. We have shown, however, that we can effectively deal with many interesting issues within the framework of this simple model. Although it is possible to extend the analysis to include, say, imperfect competition and nonconstant returns to scale, these extensions cause additional aggregation and computation problems that, at least for a first attempt, are best avoided. Finally, it would be useful to push our data base forward through 1985, so that we could attempt to model the very recent experience of the U.S. economy.14 the focus of future research. These extensions and developments will be 18 7. Data_Appendix The data required to calculate our indexes include price and quantity information on national output, capital and labor inputs, exports and imports. We have developed the output, import and export data for 1968-82 from the National Income and Product Accounts (U.S. Department of Commerce [1981],[19821,[1983fl, and have used real capital stock data constructed by the Bureau of Labor Statistics (U.S. Department of Labor [19831) and real labor data updated from Jorgenson and Fraumeni [19811, since these series closely approximate our theoretically ideal indexes. More specifically, we have calculated the value of output (ptyyt) as gross domestic business product including tenant occupied housing output, property taxes, and Federal subsidies to businesses, but excluding Federal, State and Local indirect taxes and owner occupied housing. The corresponding price index (Pt.1), was computed by cumulating the Business Gross Domestic Product Chain Price index. The value of merchandise exports and imports were determined by adding the durable and nondurable export and import values, respectively, reported in the National Accounts. Tariff revenues were added to the value of imports. For 1967-82, value and price data for nine different types of exports and ten types of imports were available, which were used to compute chain price indexes. An aggregate export price index was calculated as a translog (or Divisia in SHAZAM terminology) price index in the nine types of exports and the corresponding quantity series Xt was determined implicitly. Translog or Tornqvist indexes are defined in Diewert [19781. An aggregate nonpetroleum import price index PtM was calculated as a translog or divisia price index in the nine types of nonpetroleum imports and the corresponding quantity series 19 Mt was determined implicitly. The price index pt0 and the value series for petroleum imports were taken directly from the National Accounts and the corresponding quantity series O was determined by division. Using the values of imports and exports, ptt pt00t and PtXXt, tax adjusted gross domestic private business sales to domestic purchasers, or absorption, was calculated as PlSt = ptyt - pt)(t + pt#t + pt0t The corresponding price (pts) determined by cumulating the gross domestic purchases chain price index from the National Accounts, and the constant dollar quantity St was calculated by division. For our labor quantity series Lt, we used the series constructed by 0. Jorgenson and B. Fraumeni, whieh is conveniently tabled by the U.S. Department of Labor [1983;77]. Our total private labor compensation series, ptLLt, was taken from the same publication. The price of labor, PtL, was determined by division.15] For our capital services quantity series Kt we used the private business sector (excluding government enterprises) constant dollar capital serivces input tabled by the U.S. Department of Labor [1983;77]. In order to ensure that the value of privately produced outputs equals the value of privately utilized inputs, we determined the price of capital services PtK residually, i.e., K (Ptyyt — PtLLt)/Kt. Changing now to the notation used in the main text of our paper, we set xESt, w'2 pEPt5 ; pt0; yt3 _Lt, pt2 Xt, w3 PtL ; yt1 _Mt, w1 PtM ; yt2 _0t, The price series are tabled below in Table 8 while the quantity series are tabled in Table 9. Finally, we calculated a translog price index using P1 w1, w2, w3 as prices and xtj, yt1, yt2, yt3 as the corresponding quantity weights. Call the resulting price and net output series PAt and QAt. We calculated a translog 20 quantity index using Kt and Lt as quantities and PtK and ptL as the corresponding price weights. Call the resulting input quantity and price series QBt and PBt. We took t to be QAt/QBt, a translog productivity index. For a theoretical justification of this index, see Theorem 1 in Diewert and Morrison [1986]. The series t may be found in Table 9 below. 21 FOOTNOTES 1This concept can be traced to Samuelson [1953]. 2See Woodland [1982; 3690375] for a summary of Kohli's approach and references to related empirical literature. M. Denny and D. Burgess used this approach in their unpublished Ph.D. theses; see Denny [1972] and Burgess [1974]. 30f course, some of these export goods could be highly or even perfectly substitutable with the corresponding domestic outputs. 4See Diewert [1974] [1982], Fuss and McFadden [1978] and Jorgenson [1984] for references to the rapidly growing literature that utilizes duality theory. 5This series was taken from Jorgenson and Fraumeni [1981]. 6See Gourieroux, Holly and Monfort [1982]. Replacing B by CCt is equivalent to estimating B subject to the usual determinantal inequality conditions for positive semidefiniteness. One of these inequalities turns out to be binding in our case and this Is what leads to a failure in the usual asymptotic theory. 7All supply and demand elasticities with respect to the capital input are unity and this is due to our maintained hypothesis of constant returns to scale. We assumed a constant returns to scale technology in order to avoid conceptual problems that can occur when aggregating over producers. 8We ignore tariffs, subsidies and quotas in what follows. 9See Dornbusch [1980] for further discussion of the real exchange rate concept. 101f there were only one domestic variable good, it can be shown using the properties of profit functions that Bt'(l) must be nonpositive; see Diewert [1974;145]. -This reflects the fact that the trade elasticities tabled in the previous section tend to become larger in magnitude over the sample period. 2See Dornbusch [1985] for a nice exposition of purchasing power parity theories. 13Net capital movements abroad should be excluded. 14The main obstacle to accomplishing this task is the unavailability of a sensible measure for U.S. real labor input into the private production sector. The Bureau of Labor Statistics [1983] for historical reasons uses unweighted manhours as its official measure of labor input. The conceptually more desirable wage rate weighted series developed by Jorgenson and Fraumeni [1981] which we utilized is only available through 1982. 15We wish to thank Mike Harper at BLS and Barbara Fraumeni for their help in providing updated series. 22 TABLE 1. PARAMETER ESTIMATES FOR THE UNCONSTRAINED AND CONSTRAINED MODELS COEF. SYMBOL UNCONSTRAINED ESTIMATES STANDARD ERROR T RATIO CONSTRAINED ESTIMATES a1 3.7639 .281 13.39 3.8242 a2 -3.3242 .264 —12.57 —3.4414 a3 — .23701 .051 — 4.61 - .23835 a4 — .047639 .013 — 3.57 - .050116 a5 — .079179 .062 — 1.27 — .061502 b1 .31215 .282 1.11 .29409 b2 — .63288 .167 — 3.79 - .60521 b3 .11210 .030 3.77 .12770 b4 .036548 .012 2.96 .037821 b5 .001142 .030 .038 - .025091 b11 2.0682 7.70 .269 2.191161 b22 .065458 .030 2.19 .114062 b33 .001921 .0008 2.43 .002099 b44 - .046765 .043 — 1.08 .050206 b12 — .19818 .045 — 4.38 — .006116 b13 - .008982 .011 — .81 - .057495 .067 2.83 .556407 b14 .18578 b23 -- .004425 .002 - 2.63 — .003128 b24 .095383 .031 1.94 .035664 b34 .005532 .003 3.08 .002245 23 TABLE 2. DOMESTIC SUPPLY ELASTICITIES YEAR Ex1p1 Ex1p2 Ex1w1 1967 .615 - .550 .011 .0012 - .077 1969 .714 —.651 .016 .0013 —.081 1971 .752 - .688 .017 .0014 — .082 1973 .843 - .761 .016 .0018 — .099 1975 .883 —.753 .006 - .0007 -.135 1977 .852 — .738 .009 — .0006 —.122 1979 .943 -.809 .008 -.003 -.140 1981 .875 —.732 .010 —.017 -.136 1982 .803 -.690 .013 -.011 -.115 TABLE 3. YEAR Ex2p1 1967 Ex1w2 Ex1w3 LABOR DEMAND ELASTICITIES Ex2p2 Ex2w1 Ex2w2 Ex2w3 .894 — .911 .091 .0029 - .076 1969 1.018 —1.036 .094 .0028 —.079 1971 1.080 -1.102 .100 .0030 -.081 1973 1.155 -1.186 .120 .0037 —.093 1975 1.150 —1.191 .145 .0110 —.115 1977 1.151 —1.195 .141 .0111 —.108 1979 1.235 —1.287 .156 .0143 -.119 1981 1.162 —1.214 .148 .0214 —.118 1982 1.146 -1.191 .135 .0183 —.108 24 TABLE 4. NONPETROLEUM IMPORT DEMAND ELASTICITIES YEAR Ey1p1 Ey1p2 Ey1w1 1967 -.259 1.349 1969 —.334 1971 Ey1w2 Ey1w3 - .834 .025 —.282 1.293 — .732 .021 —.248 -.324 1.214 — .685 .019 -.224 1973 —.266 1.355 — .850 .025 —.265 1975 -.087 1.456 —1.098 .080 -.351 1977 -.128 1.310 — .958 .072 —.296 1979 -.106 1.342 -1.014 .089 —.311 1981 —.121 1.158 — .878 .121 —.281 1982 -.165 1.023 — .720 .093 -.231 TABLE 5. PETROLEUM IMPORT DEMAND ELASTICITIES YEAR Ey2p1 Ey2p2 Ey2w1 Ey2w2 Ey2w3 1967 — .286 .414 .245 —.182 -.190 1969 —.315 .424 .230 —.161 —.179 1971 -.264 .347 .188 -.130 -.141 1973 —.259 .354 .212 — .155 —.152 1975 .033 .352 .254 -.453 -.186 1977 .027 .310 .217 —.400 —.154 1979 .107 .323 .234 —.499 -.165 1981 .422 .335 .243 —.822 —.178 1982 .253 .264 .177 —.563 —.131 25 TABLE 6. EXPORT SUPPLY ELASTICITIES YEAR Ey3p1 1967 —1.626 1969 Ey3w1 Ey3w2 Ey3w3 .988 .246 .017 .375 —1.596 1.003 .229 .015 .349 1971 -1.565 .992 .226 .015 .332 1973 -1.508 .929 .235 .016 .329 1975 -1.425 .795 .242 .040 .348 1977 -1.430 .814 .240 .042 .334 1979 -1.397 .782 .238 .048 .329 1981 —1.385 .752 .230 .073 .331 1982 -1.405 .790 .224 .067 .324 Ey3p2 26 TABLE 7. IMPACT ON THE TRADE BALANCE OP A ONE CENT INCREASE YEAR ESTIMATED TRADE BALANCE t Bt(1) 1967 OlBt' (1) EQUILIBRIUM EXCHANGE RATE et 16 - .49 1.023 .37 — .51 1 .007 1 . 1968 IMPACT ON TRADE BALANCE 1969 - .30 - .55 .995 1970 — 1.80 — .62 .971 1971 -. 4.96 - 66 .925 — 7.59 — .74 .897 1972 1973 .45 -1.01 1 .004 1974 16.42 —1.68 1 .098 1975 10.86 -1.90 1 .057 -. 5.25 -1.86 .972 1977 - 9.93 -2 .08 .952 1978 -12.04 -2.36 .949 1979 - 9.53 -2.95 .968 1980 -25.28 -3.82 934 1981 -51.16 -4 .04 .873 1982 —104.05 —3.72 .720 1976 27 TABLE 8. PRICE SERIES YEAR pt1 pt2 1967 0.80182 0.72204 0.78489 0.871 0.84947 0.92252 1968 0.82588 0.77277 0.79758 0.865 0.86270 0.95383 1969 0.86056 0.82668 0.82298 0.858 0.89104 0.94054 1970 0.90359 0.88226 0.88232 0.880 0.93788 0.90078 1971 0.95239 0.93318 0.92638 0.959 0.96959 0.95611 1972 1.00000 1.00000 1.00000 1.000 1.00000 1.00000 1973 1.04900 1.06135 1.17053 1.277 1.16592 1.09631 1974 1.10873 1.15432 1.48154 4.197 1.48958 1.08115 1975 1.22404 1.22917 1.63038 4.334 1.66677 1.21930 1976 1.33298 1.32006 1.65766 4.599 1.71765 1.32665 1977 1.40762 1.40568 1.80812 4.971 1.78929 1.48218 1978 1.49912 1.53092 1.98778 4.981 1.90020 1.57942 1979 1.61455 1.66222 2.20819 7.034 2.16876 1.64533 1980 1.76632 1.78329 2.50159 11.554 2.40338 1.70086 1981 1.95356 1.93094 2.57416 12.971 2.63230 1.93726 1982 2.12937 2.06104 2.55045 12.064 2.62128 1.86782 Wi3 28 TABLE 9. QUANTITY SERIES, billions of 1972 dollars YEAR x1 yt1 yt2 yt3 Kt 1967 771.141 -543.092 —31.556 —2.4006 36.095 254.516 1.08920 1968 827.861 —556.789 —38.374 —2.7560 38.968 266.369 1.09250 1969 857.460 —576.440 —40.286 —3.0874 40.863 278.533 1.08921 1970 843.920 —570.485 —41.851 —3.3261 45.277 290.697 1.04660 1971 866.445 -573.463 —45.245 -3.8060 44.652 300.990 1.00943 1972 913.824 —595.496 —51.124 —4.6500 49.353 311.907 1.00000 1973 973.382 -625.866 -53.038 —6.5896 61.205 326.255 1.00142 1974 992.963 --630.035 -51.805 -6.3400 65.936 340.914 0.98079 1975 995.917 -611.574 -43.473 —6.2337 63.973 350.272 0.87702 1976 991.136 —635.394 —53.597 -7.5175 66.597 356.822 0.83067 1977 1072.649 -666.360 —58.176 —9.0490 66.911 366.179 0.82180 1978 1142.027 —705.067 —66.492 —8.4946 74.145 379.279 0.81093 1979 1178.245 -736.629 -67.450 —8.5985 82.617 393.627 0.78878 1980 1156.462 —743.179 —65.758 —6.8720 91.496 407.663 0.74589 1981 1180.657 -759.257 —70.896 —5.9825 88.060 419.203 0.687631 1982 1113.010 —738.415 —71.869 -5.0729 79.770 435.110 0.625447 29 REFERENCES Allen, R.G .0. 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